A $t$-all-symbol PIR code and a $t$-all-symbol batch code of dimension $k$ consist of $n$ servers storing linear combinations of $k$ linearly independent information symbols with the following recovery property: any symbol stored by a server can be recovered from $t$ pairwise disjoint subsets of servers. In the batch setting, we further require that any multiset of size $t$ of stored symbols can be recovered from $t$ disjoint subsets of servers. This framework unifies and extends several well-known code families, including one-step majority-logic decodable codes, (functional) PIR codes, and (functional) batch codes. In this paper, we determine the minimum code length for some small values of $k$ and $t$, characterize structural properties of codes attaining this optimum, and derive bounds that show the trade-offs between length, dimension, minimum distance, and $t$. In addition, we study MDS codes and the simplex code, demonstrating how these classical families fit within our framework, and establish new cases of an open conjecture from \cite{YAAKOBI2020} concerning the minimal $t$ for which the simplex code is a $t$-functional batch code.

翻译：一个维度为$k$的$t$-全符号PIR码和一个$t$-全符号批量码由$n$个服务器组成，这些服务器存储$k$个线性独立信息符号的线性组合，并具有以下恢复特性：任何服务器存储的符号都可以从$t$个两两不相交的服务器子集中恢复。在批量设置中，我们进一步要求任何大小为$t$的存储符号多重集都可以从$t$个不相交的服务器子集中恢复。该框架统一并扩展了多个众所周知的码族，包括一步多数逻辑可译码、（功能性）PIR码以及（功能性）批量码。本文中，我们确定了当$k$和$t$取某些较小值时的最小码长，刻画了达到此最优值的码的结构特性，并推导了展示长度、维度、最小距离与$t$之间权衡关系的界。此外，我们研究了MDS码和单纯形码，展示了这些经典码族如何融入我们的框架，并针对\cite{YAAKOBI2020}中一个关于单纯形码成为$t$-功能性批量码所需最小$t$值的开放猜想，建立了若干新情形。